10d1bf8c-2b8d-463c-8f68-72d077bd6def 36 4 5 decimal

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Mircea Random algorithm

This algorithm is sourced from the Mathsoft Collaboratory. The programmed version reproduces the original from author Mircea Andreacut [Roumanie]... from recollection. Random on the X & Y directions is very excellent to plot the Π canvas. The Π image is from Valery Ochov Doctor in Technology of Information [СТАРТУЕТ КОНКУРС ВИДЕОРОЛИКОВ MIP ACADEMY MOSCOW DO]. The two directions is very s l o w scale(M,Min,Max) looks the bugger that slows the computation, but by so much puzzles me as much as which way to cook beans: cook horizontal or cook vertical ? Answer: at best angle !

t n Φ 1 t n - sign + 2 / :

M Min Max scale Max Min - (M M min - M max M min - / * Min + :

n 50 :

k 4 :

k 4, 8, 16 ... ≡

Leave 'n' minimal, otherwise Smath issues: Exception of Type "SystemOutofMemoryException" was thrown

X N k n * : u 1 el 1 Random 1 el : v 1 el 1 Random 1 el : i 1 N 1 - range u i 1 + el 4 u i el * 1 u i el - (* : v i 1 + el 4 v i el * 1 v i el - (* : b i el u i el v i el Φ : 31 line for m 1 N 1 - k / 1 - range w m el b m k * j + el 2 j^* (j 0 k sum : for w 1 - 1 scale 61 line :

Y N k n * : u 1 el 1 Random 1 el : v 1 el 1 Random 1 el : i 1 N 1 - range u i 1 + el 4 u i el * 1 u i el - (* : v i 1 + el 4 v i el * 1 v i el - (* : b i el u i el v i el Φ : 31 line for m 1 N 1 - k / 1 - range w m el b m k * j + el 2 j^* (j 0 k sum : for w 1 - 1 scale 61 line :

t0 1 time :

X Y augment

1 time t0 - 0

n 5000 :

<= n = 5000, 79 sec

A much quicker X, Y random

fillRnd(n) may be customised in much simpler form.

n fillRnd i 1 n range idx i el i : for X idx n Random 0.5 - augment : 21 line i 1 n range idx i el i : for Y idx n Random 0.5 - augment : 21 line X 2 col Y 2 col augment 31 line :

Here is π demo

XY n fillRnd :

X XY 1 col :

Y XY 2 col :

t0 1 time :

Π algorithm demo UV k 1 : i k XY rows range X i el (2^Y i el (2^+ sqrt 0.5 < U i el X i el : V i el Y i el : 21 line U i el 0 : V i el 0 : 21 line if for k k 1 + : 31 line U V augment 21 line :

Very quick plot

XY 1 1 sys

Π plot,,, caliper canvas square

UV XY 2 1 sys

The irrational π is not computed from this demo, neither from the Bouffon needle. Despite the fact that many reputed Mathematicians imagined complicated formulation, NONE found the endless π, all formulations being in hidden conceptual error. The mathematical proof of π dates back 1813 from a French Géomètre [Schwab]. The irrefutable proof is the 'isopérimètres', from which the continued fraction results.

1 time t0 - 106 sec *